Experimental set up
A rectangular channel set up was used for this case study, this case
never flows in floodplain. All the information about the case of study
could be consulted here: the paper Proust
et al., (2022) and the numerical data Dataset Proust et al.,
(2024):
Here the features of the experiment:
- Length: 18 m
- Width: 3 m
- Slope : 1.04 x 10-3
- Material : glass, same friction everywhere
- Flow conditions : uniform flows with a total discharge of 14, 30,
60, 120 L/s
- Water level measurements: 161 measures from 0.50 m to 16.50 m

Laboratory measurements
Check calibration data:


First, a sample of all previously shown measurements was taken to be
used in all the tested cases. Then, a guiding question was formulated to
trigger the experiment and help provide an answer.
Calibration data

Experiment: 4_WSE_main_channel_real_uncertainty
First, a plot of the DIC criterion will be shown. Then the friction
estimations obtained with the MAP estimator, along with the simulated
WSE, will be presented for all degrees higher than 0.
DIC criterion
In this case, the best estimation indicates that the friction is
constant with any variation from upstream to downstream. That’s why, all
calibration results will be presented for this polynomial degree.

Degree 0 of the Legendre polynomial: constant friction
Check summary
Zoom into the MAP and standard deviation of the error model:
WSE in mm
| N |
2.00100e+03 |
2.00100e+03 |
2001.000000 |
2001.0000000 |
2.00100e+03 |
2001.000000 |
| Minimum |
9.54560e+01 |
2.79900e-04 |
5.498560 |
5.5250300 |
7.99571e+00 |
8.201460 |
| Maximum |
1.00168e+02 |
1.21710e-03 |
19.181100 |
16.1726000 |
2.52415e+01 |
26.604700 |
| Range |
4.71200e+00 |
9.37200e-04 |
13.682500 |
10.6476000 |
1.72458e+01 |
18.403200 |
| Mean |
9.77453e+01 |
6.29700e-04 |
10.177800 |
10.2007000 |
1.52759e+01 |
15.248600 |
| Median |
9.76870e+01 |
6.19400e-04 |
9.994480 |
10.1352000 |
1.49445e+01 |
14.868100 |
| Q10% |
9.69427e+01 |
4.68300e-04 |
7.791380 |
8.0041100 |
1.16357e+01 |
11.520900 |
| Q25% |
9.72114e+01 |
5.28900e-04 |
8.754500 |
8.8365600 |
1.33058e+01 |
12.944200 |
| Q75% |
9.82521e+01 |
7.21600e-04 |
11.360200 |
11.3442000 |
1.71028e+01 |
17.250200 |
| Q90% |
9.86340e+01 |
8.08400e-04 |
12.711000 |
12.7979000 |
1.93591e+01 |
19.604100 |
| St.Dev. |
7.16794e-01 |
1.41400e-04 |
1.978360 |
1.8320200 |
2.96707e+00 |
3.205620 |
| Variance |
5.13793e-01 |
0.00000e+00 |
3.913900 |
3.3562800 |
8.80352e+00 |
10.276000 |
| CV |
7.33330e-03 |
2.24511e-01 |
0.194380 |
0.1795970 |
1.94233e-01 |
0.210224 |
| Skewness |
4.51511e-01 |
4.02530e-01 |
0.600049 |
0.4697740 |
4.78268e-01 |
0.654463 |
| Kurtosis |
6.88676e-01 |
1.95220e-01 |
0.629770 |
-0.0279754 |
6.15238e-02 |
0.314980 |
| MaxPost |
9.75897e+01 |
6.04200e-04 |
9.236950 |
8.9984800 |
1.49100e+01 |
15.529900 |
| St.Dev. |
0.716794 |
0.141369 |
| MaxPost |
97.589700 |
0.604244 |
Estimation of the friction coefficient

Residuals
In termes of WSE

In termes of discharge

Partial conclusions
- Using only WSE observations in a rectangular channel, it is possible
to accurately and precisely estimate the friction of the hydraulic
flume’s material, assuming the friction remains constant along the
longitudinal direction.
- Convergence reached quickly during the exploration of the
parameters
- Any corelation between the parameters of the model and the error
model is detected
Degree 1 of the Legendre polynomial
Check summary
Zoom into the MAP and standard deviation of the error model:
WSE in mm
| N |
2.00100e+03 |
2001.0000000 |
2.00100e+03 |
2001.000000 |
2001.000000 |
2001.000000 |
2001.000000 |
| Minimum |
9.57859e+01 |
-6.7689700 |
2.52100e-04 |
4.846070 |
5.246050 |
7.584500 |
7.415690 |
| Maximum |
1.00246e+02 |
9.0211700 |
1.19870e-03 |
19.881100 |
19.985300 |
31.584100 |
28.704200 |
| Range |
4.46010e+00 |
15.7901000 |
9.46600e-04 |
15.035000 |
14.739200 |
23.999600 |
21.288500 |
| Mean |
9.78389e+01 |
0.8132030 |
6.46000e-04 |
10.234300 |
10.397800 |
15.231700 |
15.343000 |
| Median |
9.78244e+01 |
0.7446890 |
6.26200e-04 |
10.047400 |
10.262100 |
15.132100 |
14.950600 |
| Q10% |
9.69500e+01 |
-2.0849800 |
4.80100e-04 |
7.615360 |
7.917600 |
11.295700 |
11.402300 |
| Q25% |
9.73363e+01 |
-0.7682700 |
5.53400e-04 |
8.710490 |
8.932340 |
12.775200 |
12.912000 |
| Q75% |
9.82455e+01 |
2.4414800 |
7.22900e-04 |
11.613500 |
11.694900 |
17.279500 |
17.442100 |
| Q90% |
9.88864e+01 |
3.7013700 |
8.31000e-04 |
12.994600 |
13.005800 |
19.343300 |
19.639800 |
| St.Dev. |
7.34273e-01 |
2.3339700 |
1.37700e-04 |
2.096470 |
2.060880 |
3.222170 |
3.302210 |
| Variance |
5.39157e-01 |
5.4474300 |
0.00000e+00 |
4.395200 |
4.247230 |
10.382400 |
10.904600 |
| CV |
7.50490e-03 |
2.8701000 |
2.13231e-01 |
0.204848 |
0.198204 |
0.211544 |
0.215226 |
| Skewness |
1.47594e-01 |
-0.0940681 |
5.32159e-01 |
0.441203 |
0.614919 |
0.437301 |
0.692990 |
| Kurtosis |
1.05091e-02 |
-0.1046220 |
5.10203e-01 |
0.192198 |
0.853477 |
0.593110 |
0.764940 |
| MaxPost |
9.76659e+01 |
1.3908700 |
5.46900e-04 |
9.595060 |
9.887800 |
12.857100 |
14.110700 |
| St.Dev. |
0.734273 |
2.33397 |
0.137740 |
| MaxPost |
97.665900 |
1.39087 |
0.546857 |
Estimation of the friction coefficient

Residuals
In termes of WSE

Degree 2 of the Legendre polynomial
Check summary
Zoom into the MAP and standard deviation of the error model:
WSE in mm
| N |
2.00100e+03 |
2001.0000000 |
2001.0000000 |
2.00100e+03 |
2001.000000 |
2001.000000 |
2001.000000 |
2001.000000 |
| Minimum |
9.50853e+01 |
-7.9169300 |
-14.9179000 |
3.37500e-04 |
5.014510 |
5.386390 |
7.890460 |
7.832800 |
| Maximum |
1.00985e+02 |
8.4200700 |
17.0317000 |
1.19230e-03 |
17.935300 |
18.371300 |
24.909100 |
27.048800 |
| Range |
5.89970e+00 |
16.3370000 |
31.9496000 |
8.54700e-04 |
12.920800 |
12.984900 |
17.018600 |
19.216000 |
| Mean |
9.77880e+01 |
0.9619520 |
-0.0314878 |
6.52200e-04 |
10.205200 |
10.261400 |
15.143100 |
14.997100 |
| Median |
9.77258e+01 |
1.0716300 |
-0.3644200 |
6.51100e-04 |
9.951910 |
10.102700 |
14.854500 |
14.872700 |
| Q10% |
9.67616e+01 |
-2.1062300 |
-7.4504100 |
4.84800e-04 |
7.817560 |
7.767680 |
11.424500 |
11.469300 |
| Q25% |
9.71982e+01 |
-0.7618010 |
-4.0387600 |
5.51100e-04 |
8.602580 |
8.842100 |
13.124900 |
12.802800 |
| Q75% |
9.83237e+01 |
2.5970800 |
3.9951400 |
7.38600e-04 |
11.454400 |
11.404100 |
16.894200 |
16.714500 |
| Q90% |
9.89365e+01 |
4.2098100 |
7.5742200 |
8.35300e-04 |
12.897500 |
12.974300 |
19.012700 |
18.561900 |
| St.Dev. |
8.25961e-01 |
2.4531700 |
5.7153800 |
1.41500e-04 |
2.071270 |
2.103930 |
2.993670 |
2.851680 |
| Variance |
6.82211e-01 |
6.0180500 |
32.6656000 |
0.00000e+00 |
4.290140 |
4.426520 |
8.962030 |
8.132060 |
| CV |
8.44640e-03 |
2.5502000 |
181.5110000 |
2.16982e-01 |
0.202961 |
0.205033 |
0.197692 |
0.190149 |
| Skewness |
2.81157e-01 |
-0.0207898 |
0.1763540 |
4.60201e-01 |
0.643372 |
0.683571 |
0.462104 |
0.419420 |
| Kurtosis |
1.79208e-01 |
-0.1019750 |
-0.2726330 |
3.71729e-01 |
0.453064 |
0.844478 |
0.223736 |
0.460300 |
| MaxPost |
9.76545e+01 |
0.6991390 |
0.1643870 |
6.38800e-04 |
9.070160 |
9.682910 |
14.301500 |
14.518400 |
| St.Dev. |
0.825961 |
2.453170 |
5.715380 |
0.141518 |
| MaxPost |
97.654500 |
0.699139 |
0.164387 |
0.638756 |
Estimation of the friction coefficient

Residuals
In termes of WSE

Degree 3 of the Legendre polynomial
Check summary
Zoom into the MAP and standard deviation of the error model:
WSE in mm
| N |
2.00100e+03 |
2001.0000000 |
2001.0000000 |
2.00100e+03 |
2001.000000 |
2001.000000 |
2001.000000 |
2001.000000 |
| Minimum |
9.50853e+01 |
-7.9169300 |
-14.9179000 |
3.37500e-04 |
5.014510 |
5.386390 |
7.890460 |
7.832800 |
| Maximum |
1.00985e+02 |
8.4200700 |
17.0317000 |
1.19230e-03 |
17.935300 |
18.371300 |
24.909100 |
27.048800 |
| Range |
5.89970e+00 |
16.3370000 |
31.9496000 |
8.54700e-04 |
12.920800 |
12.984900 |
17.018600 |
19.216000 |
| Mean |
9.77880e+01 |
0.9619520 |
-0.0314878 |
6.52200e-04 |
10.205200 |
10.261400 |
15.143100 |
14.997100 |
| Median |
9.77258e+01 |
1.0716300 |
-0.3644200 |
6.51100e-04 |
9.951910 |
10.102700 |
14.854500 |
14.872700 |
| Q10% |
9.67616e+01 |
-2.1062300 |
-7.4504100 |
4.84800e-04 |
7.817560 |
7.767680 |
11.424500 |
11.469300 |
| Q25% |
9.71982e+01 |
-0.7618010 |
-4.0387600 |
5.51100e-04 |
8.602580 |
8.842100 |
13.124900 |
12.802800 |
| Q75% |
9.83237e+01 |
2.5970800 |
3.9951400 |
7.38600e-04 |
11.454400 |
11.404100 |
16.894200 |
16.714500 |
| Q90% |
9.89365e+01 |
4.2098100 |
7.5742200 |
8.35300e-04 |
12.897500 |
12.974300 |
19.012700 |
18.561900 |
| St.Dev. |
8.25961e-01 |
2.4531700 |
5.7153800 |
1.41500e-04 |
2.071270 |
2.103930 |
2.993670 |
2.851680 |
| Variance |
6.82211e-01 |
6.0180500 |
32.6656000 |
0.00000e+00 |
4.290140 |
4.426520 |
8.962030 |
8.132060 |
| CV |
8.44640e-03 |
2.5502000 |
181.5110000 |
2.16982e-01 |
0.202961 |
0.205033 |
0.197692 |
0.190149 |
| Skewness |
2.81157e-01 |
-0.0207898 |
0.1763540 |
4.60201e-01 |
0.643372 |
0.683571 |
0.462104 |
0.419420 |
| Kurtosis |
1.79208e-01 |
-0.1019750 |
-0.2726330 |
3.71729e-01 |
0.453064 |
0.844478 |
0.223736 |
0.460300 |
| MaxPost |
9.76545e+01 |
0.6991390 |
0.1643870 |
6.38800e-04 |
9.070160 |
9.682910 |
14.301500 |
14.518400 |
| St.Dev. |
0.825961 |
2.453170 |
5.715380 |
0.141518 |
2.07127 |
| MaxPost |
97.654500 |
0.699139 |
0.164387 |
0.638756 |
9.07016 |
Estimation of the friction coefficient

Residuals
In termes of WSE
